WORDS AND AND

995

591

INCLUSIONS IN ELASTO-PLASTIC SOLIDS UNDER THE INFLUENCE OF INTERNAL AND EXTERNAL PRESSURES

TAPAN KUMAR DAS

IndianInstituteofMechanicsof Continua 201 ManicktolaMainRoad, SuiteNo.42

Calcutta- 700054,India P.R. SENGUPTA

Department

ofMathematics University of Kalyani Kalyani,

West Bengal,

India

and LOKENATHDEBNATH

Department

of Mathematics Universityof CentralFlorida Orlando,

FL

32816,

U.

S. A.

(Received Febrt,arv 3, 1992 :rd it revised form April 10, 1993)

ABSTRACT.

An

attemptismadetostudytheproblems^ofsphericaland circular inclusions in elasto- plasticsolidsunderthe action of internalincreasing

pressure

and externalconstant

pressure,

takinginto considerationofthework-hardening effect. Particular attention isgiventothe linear workhardening effectonboth

problems.

Itisshownthatresultsofthisanalysisare ingood agreement^withthose of idealplasticsolids.

KEY WORDS AND PHRASE.

Sphericaland circular inclusions,elasto-plasticsolids.

1991

AMS

subject classification codes: 73E.

1.

IN’IODUCTION.

Severalauthorsincluding

Mott

and Nabarro ],Eshelby

[2

5],

Jaswon

andBhargava

[6]

Willis [7

8]

have made systematic investigations oftheproblems of sphericalandcylindric’,dinclusions in an elastic medium. BothBhargava

[9]

and

Sengupta [10]

haveusedtheprincipleof minimum

energy

^to

study problems

of symmetric anisotropicinclusions in anelastic medium.

Tresca

11 has shownthat a metal yields plastically whenthe maximum shearstress attains acritical value.

Bhargava [12]

has investigatedtheproblemsof inclusions inelasto-plasticsolidsunder theassumptions ofthe infinitesimal theory ofstrainandperfectplasticity conditionssatisfiedbyanelastic-plasticsolid. Fromaphysical point of view, thework-hardeningeffectisimportantin aplasticmaterialeventhoughthe

consequences

of this effect arequite complex. Hopkins 13]has included thework-hardeningeffectof the material in

592 ’1. K. I)AS, P. R. SI..NGIlP’I,\ ,\NI 1.. I)ICBNA[’lt

his

problem

of dynamical expansion of sphericalcavities in metals.

Sengupta

and his associates

[14 15]

have studiedthe

problems

ofinclusions inelastic-plasticsolidsof work-hardeningmaterialoffinite and infiniteextent.

On

the otherhand, Tokuoka

[16]

has investigated the

plastic

deformations and instability ofspherical shells under internal

pressure. In

spiteof thisprogress,^furtherinvestigationof the inclusionproblemsin anelastic-plastic mediumisneeded.

Thispaperisconcernedwiththeproblemsofsphericaland circular inclusionsinelasto-plastic solids under the action of internal increasing

pressure

and external constant

pressure,

taking into considerationof thework-hardening effect. Special attention is givento the linearwork-hardening

effect

onboth

problems.

2.

SPHERICAL INCLUSION IN ELASTO-PLASTIC SOLIDS.

We

consider a spherical region ofradius a in a finiteelasto-plastic material which tends to undergoadimensionalchangetoasphereof radius a(1 +

i)

where lies inthe region of plasticstrain.

We

designate the spherical region asinclusion,and theoutermaterialas^matrix,where its external boundary isasphericalsurface of radius b. Thesphereisunderconstantexternal

pressure,

because of theconstraintsofmatrix, stresses

appear

both in the inclusion and the matrix. Hill 17]considered the problem^ofsphericalshellunderuniform

pressure

on itscavity surfaceundertheassumptionsof finite plastic^strains.

We

assume that the internalboundary ofthe matrix

(a

sphere of radius

a)

isundera

gradual

increasing

pressure p. At

firstatmoderate

pressure p,

thematrixbehaves likeanelastic material.

Introducing spherial polarco-ordinates(r, 0,

)

and thecorresponding displacement components (u,^v, w), for radial symmetry of deformation we

suppose

u=u(r), v=0 and w=0

(2.1)

In

absence ofbodyforce, theonly non-vanishing equation of equilibriumsatisfiedbythedisplacement component uis

+-- ^{=o. (2.}

The solutionofthe above differentialequationis u

Ar + B

r2, (2.3)

where

A

and

B

areconstants. The stress components are,therefore,

o"

(3;1. + 2)

4p

-y, B ^(2.4ab)

fro=o, =(3) ⁺ 2/)+ ^2/ 7. B

AccordingtoHenckyand^yonMises,the

yielding

^{commenceswhenthe}maximumof

Io’ ^o’,1

^reachesa

critical value

v,

where

v

isa material constant.

Now o’ rr, 6t B

^ismaximum when r a. Therefore,theyielding commencesatr a and the

corresponding pressure

_P0isdetermined

by

the followingconditions

I ^]

[ar],=a

=--P0’

[r]__ ^=-p,, 6P7 ^{7 (2.5abc)}

where P0

>>

P,.

Thus,

yielding

begins when

IN;IL:51t)NS IN t’.I.,\ST()-II,\SI IC b)ll!)S 593

Po 7’

+

p

(2.6)

andthecorrespondingdisplacement^atthe innerboundaryisgivenby

2( a_ _/

^a

U (a)

- ^?’a

^3,

^{+ 2,u}

^-7b

^{+ 3& +}

^2m

^(2.7)

Withincreasing

pressure

aplastic region spreads intotheshell. The plastic

boundary

bea

spherical

surface, its radiusat

any

moment is denoted

by

c.

Thestressesanddisplacementin the elasticregion

(b ^3-1 / (a3

^r-

/

o’,

=-A +B ^--1,

c

< ?’ <

^{bare stillofthe foma}

cr

o=O’=A

⁺¹

-B

^{+1 (2.8abc)}

b

-

u=At 3r+2/ r+4// ^-B r+3z+2// 4//

intheregion c<r

<

b.

We

mustconsidertheplasticsolid in theregion a<r<c

0o’,

2

+ 7(err ^tro)

⁰

subjecttotheyieldscondition

tro-

^o’,

?’.

Solvingthe differentialequationandusingthe condition ofcontinuityof thenormalstress atthe elasto- plasticinterface r c, we have

or, _2?’logC ^r- 2( c3)

^1-

^-p,

o=,=?’_2rlogC_

^r

^--?’

²

(c33)

^1-

^{-p, (2.10ab)}

If

o-

^{-p, at r a, then}

p

2?’log^c

+

1-

+ p[

a

(2.11)

From

the condition intheplastic region

e ⁺ 2eoo -E2 ^{v (O’ + 20"o)}

where

v

and

E

arethe Poisson’s ratio and

Young’s

modulusrespectively,and the conditionof continuity of displacementatthe interface, the radialdisplacementuintheplastic region a

<

r

<

cis

_rr ^27"c .41.tc3_+3_Kb )

^c

^{27’(1 2v)} [

_log-

^c-lc3 1]

--3-+

^r

u=3K

b, ^12,u -3"+ ^-3"+

r E r

3 ^{-P" (2.12)}

Thedisplacementatinnerboundary^is

I( ) ] ^[c-1c31]

^a

27’c3 ^41.tc3+3Kb

^lc

^27’(1-2V)

log--

ff-+ ^a-p,

u(a)=- Lt ^{i -fi} ’-’’+

^a

^E

^a

⁷

We

know thattheradialdisplacement ofthe inclusion

isa(8- e),

^and

a and therefore _pl

3K’(- e) u’(a) p

3K-

(2.13)

Now by usingtheconditionofcontinuityof normalstressatthe inclusion andmatrix,we get

3K’(8 e) 2?’logC +

1-

+ (2.14)

Thedisplacementof the innerboundaryof themau’xisawhichby equation (2.13)^is

ae c-

3K

]2kt b75-3 -5

^{+ a E log +3}

3

^{-5- (2.15)}

Therefore

= (1-

^Y

^{C3 v)}

^(2.16)

The relation

tween

eand isgivenby

-+--- P _(2.17)

a 3 3

3K"

The material in the region a g r gc is under elasto-plastic strainand satisifes the work-hardening conditions. Ifwedefine

e;, e;

^{as the radial} and tangentialelastic sains,

e, e

^{asthe radial and}

tangential plastic strainsand

e,eo

asthe coespondingquantitiesof total sain and if u the radial displacement, then we have thefollowingrelations

e e ^{+ e,}

^e

^{e; +} e ^(2.18)

Ou

u

e , ^e _=-r ^(2.19)

E e

^2Vo,

Ee ^{(1 V)} V ^(2.20)

It

is

customa

^toassume thattheplasticsain satisfies theincompressibilitycondition

e ⁺ 2e

^0.

^(2.21)

Then the following compressibility can derivedbyusingthe equations(2.18) (2.21)

+ 2 _0r ^{+ (2.22)}

where

K

isthebulk modulus.

The elasto-plasticmaterial safifiesthe following

equilibrium

equation

O

2

a, =7 ^{(a a’) (2.23)}

The

sess-sn

ce forawork-hdening materialinuni-axialcompressionisof the fo(sHill [J])

r ^{+ H(e) (2.24)}

where

,

earecompressive

sess

andstrain(both

ten

aspositive),

T

the initialyield

sss, H

is

e

degree of hardening expressed as a function of total strain. Evidently, in radially symmeic deformations,

y

elementof material is

subject

to a uni-axi, radial

compressive sss

state

- ^together

^withahy&ostatictensile stress

a.

^{Th latter}

^{sess, by}

^itself,

^produces

^{a posive}

isotopic,elastic sain ofamount

{(1- 2v)a}

^/

^E.

Rememberingthesignconventionfor

8

de,

e

appropfiageneralyieldcriterionis

o-, y+H{-e,+ ^(1- 2v)}E ^(2.25)

If

ere

^isnoBauschinger effect, thenH(e)=-H(-e),i.e.,

H

(e) isan odd functionof

strn.

Thus

e gener

yieldcriterion fora work-hdeningmatefi is

fro ff + H -e ⁺ _E

¹

^(2.26)

In

caseoflinework-hardening,

H

is a linear functionoftotalsain andan analytical discussionis possible.

In

such acase therateof work-hdeningis constantand wesupposethe yield cri:eNen as

+ffo

^(2.27)

where

H’(e)= E,

gradient of thestress-strain curveinthe plasticrange, supposedconstant. Solving (2.22)and(2.27),^wefind

cr

o’

(1- ^E,

³

(2.28ab)

Eliminatingthestresses

o’

^and0" between theequations

(2.23)

and(2.28ab)we obtain thefollowing ordinarydifferentialequationfor u.

d

(d--r ⁺ 27u ^{) 6y(1--/1 (2.29)}

dr

(3K + E,)

^r

The solutionof the differentialequation(2.29)is

U

A,r ++r ^?’

^3K

⁺ E, ^r(31ogr- 1) ^(2.30)

where

A

^and

B

^{arearbitraryconstants.} Therefore, with thehelpofboundaryconditions

(2.5abc)

and the continuity condition, the stresscomponents and displacementcan be obtained from

(2.30)

and

(2.28ab)

inthe plastic region a_<_r

<

cand theyare

2y(1---)log(_)3 ^{4?’ E,(1-v)+4’ E,(1-v)}

O’= 3(1+3__)

³

E(I+-)

³

E(I+E__)()

^-Pl,

1-v

](C3r3 ^{c3a c3 Pl}

+)

^log

^2, +J ^{5 )+}

²

-_v

3KE(b-c)-E’(Ec+ 3Kb) +(1-@)log

^r+

E(I+)

"=-( E(ll ⁺ )

^3KEb

::[3bc(4-3K)(1-5){4E, r+9Ka(1-5)} ],

and

(2.31)

where

590 I. Ix. I)AS, P. R. SENG[;I)[.\ AND l,. I)EBN,\’It]

q

=9Ka

1--- 12Kb3(E,-31a)+121uc’(E, +3K)]-4E,[3Kb(_K+411)]. ^(2.32)

Therefore, thedisplacementattheinternalboundary ofthematrixis

L ^{3 +}

¹⁺

^logTa

)’(1E(1 ^{-v)(1- + _) E--)c3}

^a

^{P" q [ab3c3(4 3K)(1- ){4E,}

^a

^{+ 9Ka(I (2.33)}

It

isnotedthatthisdisplacementinduces the initial elasticpart of thedisplacements.

We

considertheinclusion which isunder a normal

pressure p

onits

external

boundary andis therefore

< ^{-p,, a; -p,, r;o o.}

Itisnotvery^difficulttomark with thehelp oftheyieldcriteria of

Tresca

11 that the materialof the inclusion neveryieldsandit isalwaysinastateof elastic deformation. Thedisplacementatthesurface of theinclusion isgivenby

u’(a)= ^{ap( (2.34)}

3K’

By

usingthe conditionof continuity ofnormalstress atthe inclusionand matrixboundary, we get

E

(_) ^{4?’ E,(I-v)c}

3K’(d;-e)= ^21’ 1----

_l+31og

₊

E,

a

E,

₃

E(1

⁺

_3__)

3 3K

-2’cA+-

^b

^(2.35)

Thedisplacementof theinnerboundaryofthe matrix will be ae,whichby equation

(2.33)

is

=-(--3 _E

^1--2V

[3KE(b3-c3)-E,(Ec3+3Kb3) ⁽⁺ 1--/logC3]a

t 1+- ⁾

P[g663c3(4-3K)(1-){4E, a+9Ka(1-)}]

E 1+

(2.36)

Solving ford; with the aid ofequations

(2.35)

and

(2.36),

weobtain

-v) c3[1+E,(

⁴

(2.37)

Ifthebulk modulus of thematrixand the inclusion are same, then

t ^y(1- v)c’ ^p, bc3(4u 3K’)(E,- 9K’)

E

a

3K’ q ^(2.38)

where

INCI.USIt)NS IN t.I.ASIt)-I’I.\SI 1(; SOI,IDS _597

q

12a3(9K ^’- E,)[K’b’(E,- 3)+ lIc’(E, ⁺ 3K’)]- ^{12E, K’b(3. + 2//)}

This result is essentially the same ifthe work-hardening effect of the plastic material ^is altogether absent.

Now for theelasto-plasticsolidsof work-hardeningmaterial,the relation between e and

t

is gven by

27

^1-

E,

() --(l+31og.) ^{41,’ E,(1-v)} (.):+2Y(c_)

e=- I+--3KE’ ^9KE(I+)

^9K,b)

q

Forapicularly plasticmaterial,

e

corresponding resultisgivenby

e log-

+ (2.40)

3Kk

^{a 3 3}

Comping theabove results itfollowsthat the relation betweene and for thecase of sameinclusion andthe

maNx

materialdepends

(i)onPoisson’s ratio

v, Young’s

modulus

E,

bulkmodulus

K’

of the inclusionandontheyield

sss

in case of perfectly plastic solid, and (ii) ontherate ofconstant work-hdeningfactor

E,

^sides

qutitiesalreadymentionedin(i).

Theequilibriumpressureof the inclusioninthe present case isgivenby

(2.31),

whichmay

comp

to

uation

^{(2.11)for the}perfectly plasticcase.

Another impoant pointinthepresentcase isthejumpinthe

hoop sess

on

e surface

inclusiongivenby

31-@ -+ ^Yc + ^3p, (2.41)

+

a

Thisresult is independent of ande butdependsontheconstant teofwork-hdeningfactor

In e

case cf

perfectly

plasticsolids thejumpin thehoop

sess

is%^theyieldstss,whichcan duced

om

theaboveresult by putting

E,

^0.

All

oer

impoant results involved in

(2.31), (2.33)

and

(2.39)

are found to

agree

with

coesponding

resultsfor

pefftly

plastic solids given by

Bhgava 12],

if weput b

, p,

0 d

E,

^0,inthe above

uations.

3.

CIRCULAR INCLUSION UNDER CONDIONS OF PLA SAIN

Bhargava 12]discussedtheproblemofcircular inclusion inanelasto-plasticmaterial of infinite extent.

We

consdierhere circular cylindrical inclusion in a finiteelasto-plasticmaterialwitheffects of work-hardening. The cylinderisunderconstantexternal

pressure p,. As

the

problem

ofcircular cylindrical inclusioneven under theconditionsofthe

plane

strain is muchmoredifficultthan that of sphericalinclusion, wesuppose v for compressiblematerial. Thisassumption greatly simplifies the

2 solutionoftheproblem.

Introducingthecylindricalcoordinates

(r,O,z)

and the corresponding displacements (u,v, w), weassume for the presentproblem

u=u(r),v w

O"

Proceeding in asimilarmanneras the case of spherical^inclusion, wesupposethattheinternalcircular boundaryof the matrixis underapressurep within the elasticlimit. Ourdiscussionisconfinedtothe scopeof the infinitetheoryof strain.

The

only

equationofequilibriumunder nobodyforce is

+

=0

(3.1

dr k,dr and hence the solutionis

u=Cr+-- D

(3.2)

t"

whereCandDarctwoarbitraryconstants.

Theyieldingcommences when the maximum of

Io’o o’rl

^{attains a}critical value ?, where],isa materialconstant.

Now o’o ^o, 4// D

^{ismaximumwhen ,"} a. Thereforethe yielding begins^atr a and the correspondingpressure

Po

is determinedby thefollowingconditions

[(Tr]r=a

^=--Po,

[(Tr]r=o=--Pe

and

7 4//-y D

^(3.4)

where P0

>> Pe

Therefore yieldings commenceswhen

p0

- ^{+ p (3.5)}

andatthisstage thedisplacement of theinternalboundaryis

Uo(a)=[b ^{(A. +//)}

^a

⁺ -] ^{2( a2 +//)}

^pe

^(3.6)

For

anincreasing

pressure beyond 1--- ^{+ p,}

^{theplasticzone is}

^developed

ⁱⁿthe matrix,andif c be theradiusof elasto-plastic interface, then proceeding as inthe case of spherical inclusion under external

pressure,

theelastic stressesand displacementinthematrix

beyond

c aregiven

by

(r,,

c - ^p,,

^a

^{-c 7’ +}

^{p, o}

^{v(o- +}

^o-

^o)

^v

^{2p, (3.7abe)}

and

"=

4

b( +//)

^r

⁺ 7 2(. +//) ^{p (3.8)}

The material in the region a_<r_{_< c is}elasto-plastic and satisfies the work-hardeningcondition.

Presentingananalysissimilartothatofasphericalinclusion, theequilibrium equationis

--- 0o’r ⁺

^r

^{(o’ ’)}

⁰

^(3.9)

and the yield criterion of linear work-hardeningmaterial is

Theplasticmaterial satisfiesthe compressibility equation

Ou

u

Or

r

(3.10)

(3.11)

Sincethe displacementcomponent u is continuousonthe elasto-plasticinterface r=c,thenit isgivenby

4

b"(& +p)c+

^r

2(; ⁺ p)

^(3.12)

Eliminatinguando" fromequations (3.9),

(3.10)

and(3.12),weobtain

rr -r-

^1-

^{+ E,} -c b2()+p)

^c

⁺

c jr

^E, 2(A. +p),

^"3

^(3.13)

Solvingthe above differentialequationfor the normal stressandusingthecontinuitycondition onthe elasto-plasticinterface r=c we have

o’, 7’log-

+

r

b2(2 +

2

=

(3.14ab)

2

7’

^1- 1-2log

+cZ ^E,

-C b2(+,f/)c+v ^{+ V- -P"} 4(;/.t)+1

given by

-Pl o -Pl z

^-2

^{VPN1 (3.17ab)}

u’ (1 ⁺ v’)(1- 2v’)

-p (3.18)

r

E’

where

E"

and

v"

aretheYoung’smodulus and Poisson’sratiorespectively.

According to Bhargava [9] the inclusion never yieldsfor infinitesimal strains. The circular inclusion of radius of radius a spontaneously undergoes dimensional change to a circle of radius

a(1 ⁺ )

inthe absenceof matrix and attains the radius

a(1 ⁺ e),

when it is inequilibriuminthe

presence

of thematrix. Presentingananalysissimilartothatof asphericalinclusion, we obtain

]1 ^Pe

:=4 kb2(;I,+//)c+-

^a

^2(+//)

(1 + V’)(1-

²

v’) E"

(3.19) (3.20)

where Pl^isgivenby (3.16).

Theseresults

(3.15)

(3.20)for elasto-plastic solidswithwork-hardeningeffects are ingoodagreement withthe correspondingresults of aperfectly plastic solid if we put _1), 0 and

E,

^{0 in the above}

results.

The

pressure

_Plattheinternal

boundary

ofthe matrix is

" @_) ^7’ a2[ff_ { l}(1 ¹⁾⁽¹ cl__/]

p,:il- 7’log- b2(X+/---c’+-- ^{+ F 5---- E,c +} -

As

regdsthe inclusion, if itisunder unifopressure

p,

thepfincipNstressanddisplacementfield is

The jumpinthehoopstressaswecross thecommonboundaryof the inclusion and the matrix is

60

o0

^1-

⁺

9( ⁺ flc ^{+ p, (3.21)}

If

E, O,

thisresult reducestothatof

perfectly

elasticsolid.

It

isimportanttopointoutthatin case ofperfectlyplasticsolidthehoop suessisindependent of the external boundary of the^matrix.

Finally, all the results obtained in the above analysis depend on the constant rate of work- hardening factor

E,

whichplaysanimportant roleinplasticity.

REFERENCES

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

Mott,

N.

F.,and Nabarro,

F.

R. N.,

Proc. Roy. Soc. A

52

(1940)

p. 25.

Eshelby,

J.

D.,Phil,

Trans. Roy

^Soc,

A. 244 (1951)

p. 87.

Eshelby, J. D.,

Prof. Roy. Soc.

^(London)

.A.

^241,376(1957) p. 376.

EshelbyJ. D.,

Proc. Roy. Soc.

(London) A.

252

(1959b) p.561.

Eshelby, J. D.,

Progress

insolidMech:ni;Vol.II (Ed"Sneddon and Hill)

(1961).

Jawson,M. A.,andBhargava, R. D., PrQc.

Carnie.

Phil..

Soc.

57 (1961) p. 669.

Willis,J. R., Jo0r,Mech.Ph.y,Solids

V91. 1

(1965) p. 377.

Willis,

J.

R.,

Ouart. Jour.

Mech. Appld. Math..Vol.17

(1964) p.

156.

Bhargava, R.D., Appl.

Sci.

Res A,

Vol.

II, (1962) p.

80.

Sengupta, P. R.,

Ind.

J.

Theo. Phys. Vol.

XI, No. (1963) pp.

27-37.

Tresca,H., Compl,Rend.59

(1864)

p. 754.

Bhargava, R. D.,Ind.

Jour.

M0.1;h, Vol. 5,No.2(1963)

Hopkins,H. G.,

Progress

insolid^Mechanics,Vol. (Ed:Sneddon and Hill)(1961) p. 83.

Sengupta,

P. R.,Ind.

J.

Mech. Math.

Part II,

Specialissue,

(1969)

pp. 80 89.

Sengupta,

P.

R.,

and Kundu,

H. N., InO, J,

The0,Phys.Vol. 18,

No.

4

(1970) pp.

109 121.

Toku ka,Tatsuo,Indiana Univ.,l,20 (1970/71) pp. 193 206.

Hill,

R.,

The Mathematical^theoryofolasticitv,OxfordUniv.

Press (1950).